Introduction to Bernoulli’s equation and It’s Application
The Bernoulli’s equation is one of the most useful equations that is applied in a wide variety of fluid flow related problems. This equation can be derived in different ways, e.g. by integrating Euler’s equation along a streamline, by applying first and second laws of thermodynamics to steady, irrotational, inviscid and in-compressible flows etc. In simple form the Bernoulli’s equation relates the pressure, velocity and elevation between any two points in the flow field. It is a scalar equation and is given by:
|Basic Bernoulli’s Equation|
Each term in the above equation has dimensions of length (i.e., meters in SI units) hence these terms are called as pressure head, velocity head, static head and total heads respectively. Bernoulli’s equation can also be written in terms of pressures (i.e.,Pascals in SI units) as:
|Bernoulli’s equation in terms Of Pressure|
Bernoulli’s equation is valid between any two points in the flow field when the flow is steady, irrotational, in-viscid and incompressible. The equation is valid along a streamline for rotational, steady and incompressible flows. Between any two points 1 and 2 in the flow field for irrotational flows, the Bernoulli’s equation is written as:
|Bernoulli’s equation with Datum|
Bernoulli’s equation can also be considered to be an alternate statement of conservation of energy (1st
law of thermodynamics).
The equation also implies the possibility of conversion of one form of pressure into other. For example, neglecting the pressure changes due to datum, it can be concluded from Bernoulli’s equation that the static pressure rises in the direction of flow in a diffuser while it drops in the direction of flow in case of nozzle due to conversion of velocity pressure into static pressure and vice versa. Figure 1 shows the variation of total, static and velocity pressure for steady, incompressible and inviscid, fluid flow through a pipe of uniform cross-section.
Since all real fluids have finite viscosity, i.e. in all actual fluid flows, some energy will be lost in overcoming friction. This is referred to as head loss, i.e. if the fluid were to rise in a vertical pipe it will rise to a lower height than predicted by Bernoulli’s equation. The head loss will cause the pressure to decrease in the flow
direction. If the head loss is denoted by H then Bernoulli’s equation can be modified to:
|Fig 1. Application Of Bernoulli Equation.|
Figure 1 shows the variation of total, static and velocity pressure for steady, incompressible fluid flow through a pipe of uniform cross-section without viscous effects (solid line) and with viscous effects (dashed lines).